TECHNIQUES FOR DECISION MAKING IN RISKY CONDITIONS

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1 RISK AND UNCERTAINTY THREE ALTERNATIVE STATES OF INFORMATION CERTAINTY - where the decision maker is perfectly informed in advance about the outcome of their decisions. For each decision there is only one possible outcome which is known to the decision maker. RISK in this situation a decision may have more than one possible outcome, so that certainty no longer exists. However, the decision maker is aware of all possible outcomes and knows the probability of each occurring. UNCERTAINTY In this situation a decision may have more than one outcome and the decision maker does not know the precise nature of these outcomes, nor can they objectively assign a probability to the outcome. TECHNIQUES FOR DECISION MAKING IN RISKY CONDITIONS If the decision maker knows the possible outcomes that may result from a decision and can assign probabilities to each of those outcomes, then the expected monetary values may be substituted for certain values in choosing between alternative courses of action. The expected monetary value (EMV) of a particular course of action may be defined as: EMV P i V i Where P i = the probability of the i th outcome V i = the value of the i th outcome P i = 1 EMV is the weighted sum of the possible outcomes, when each outcome is weighted by its probability, and all possible outcomes are taken into account.

2 Example: An ice cream shop may know that its takings vary with the weather, which may be sunny [with a probability (p) of 0.2] or cloudy (p = 0.4) or raining (p = 0.4). In this case the EMV is calculated as shown below: EMV = $500 (0.2) + $300 (0.4) + $100 (0.4) = $260 In the example given here there are only three possible weather conditions and three probabilities, which sum to one (1). A discrete probability distribution If the distribution of ice cream takings is characterized by a normal distribution (continuous), then the EMV of the takings is given by the mean of the distribution.

3 A continuous probability distribution LIMITATIONS OF THE EXPECTED VALUES a) If the expected monetary values are used as the decision criterion, then the rational decision-maker deciding between two alternatives will always choose the course of action that yields the highest EMV. However while this may appear an intuitively sensible way in which to take decisions, a number of examples show that its application can lead to a number of quite nonsensical conclusions. Imagine that a rational person is asked to take part in a game with another player which consists of tossing a coin for a stake of $1. If the coin lands head up there is a gain of $1. If the coin lands tails up there is a loss of $1. Provided that the coin is a fair one, so that heads and tails are equally likely, the expected value of this game is equal to: 0.5 ($1) 0.5 ($1) = 0. A person using the EMV criterion would be absolutely indifferent to whether they played the game or not. The slightest inducement to play (a bribe of one cent, for instance) would be sufficient to encourage them to take part. When the stakes are small this analysis seems intuitively acceptable. However, the expected monetary value of the game remains exactly the same as the stakes rise. If the stakes were $ the expected value would still be zero and a bribe of one penny would be sufficient to encourage a rational person using EMV as the basis for the decision to take part. This seems intuitively much less plausible, as it seems highly unlikely that a rational person would be prepared to risk losing $ , even if that possibility were offset by the equally likely prospect of winning $ In everyday language it seems sensible to

4 suppose that most people would care more about the prospective loss than the prospective gain. b) St Pertersburg Paradox makes the same point clear in an even more dramatic way. Consider the situation where a coin is tossed and a payment is made to the player depending upon which toss of the coin first comes up heads. If it comes up heads first time the payment is $2. If it does not come up heads until the second toss, the payment is $2 2 = $4, and if it does not come up until the n th toss, the payment is $2n. How much would a rational person be willing to pay to take part in this game? The expected monetary value of the game is given by: EMV = 0.5 (2) + (0.5 2 ) (2 2 ) + (0.5 3 )(2 3 ) (0.5 n )(2 n ) EMV = EMV = INFINITY In other words the EMV is infinity, and a person using the EMV as a means of decision-making would be willing to pay everything they have to take part in the game. These examples illustrate a major problem with EMVs as a means of taking decisions. Individuals will not accept fair bets involving large amounts of money because they CARE more about the possibility of a loss than they do about the possibility of an equal gain. In the language of economists the UTILITY lost as a result of losing $100 may be more than the utility gained by winning $100. UTILITY AND ATTITUDES TOWARDS RISK The analysis above suggests that EMV has serious limitations as a criterion on which to base decisions. It seems intuitively likely that the value or utility placed on a loss of $100 by a rational individual may well exceed the utility arising from a gain of $100. That suggests that a explicit examination of the links between utility and income may help to provide an alternative means of assessing decisions in situations of risk. Figure below shows three different possible relationships between an individual s level of income and the utility they experience as a result of having that income. Each can be seen to illustrate different attitudes to risk on the part of the individual concerned. In figure (a) the curve linking utility and income becomes less and less steep at higher levels of income indicating decreasing marginal utility of income. If such an individual is at level of income A, which

5 gives them utility X, and is considering whether or not to accept a fair bet with a 50/50 probability of either increasing their income to C or reducing it by an equal amount to B, they will consider the impact on their utility. If their income is increased to C their utility rises to Z. On the other hand, if their income is reduced to B, their utility falls to Y. However, as figure (a) clearly shows the decrease from X to Y is substantially larger than the increase from X to Z. If these are equally likely as in the example given, the individual in question will not accept the bet offered. The type of behaviour shown above is known as RISK AVERSE behaviour, and it is frequently assumed that most individuals and companies behave in this way. In figure below the link between utility and income is drawn as a straight line. In this case, the individual places exactly the same value on a loss as on a gain of the same monetary value, and is described as RISK NEUTRAL.

6 For individuals whose preferences conform to this relationship, EMVs are an appropriate reflection of their decision-making. It can be seen, then, that EMVs represent a special case within the general framework of varying attitudes to risk. Figure c) illustrates the relationship between income and utility for a RISK LOVER. For this individual the utility attaching to the gain in income from A to C is clearly larger than that arising from the loss from A to B. Such an individual will accept fair bets, even for large amounts. In each of the examples given above the criterion of EMV has been replaced by that of expected utility (EU). Instead of choosing whichever course of action offers the highest EMV, the decision-maker chooses that which gives the highest EU, where:

7 EU = P i U i where P i = probability of the i th outcome U i = utility of the i th outcome and P i = 1. The adoption of the expected utility criterion provides a means by which different attitudes to risk can be taken into account when modeling decisions. However it can be difficult to use it in a normative way, as a prescription and a practical tool, rather than as a means of predicting the firms behaviour in a general way. That is because it involves estimating the relationship between utility and income for a particular decision-maker (or for the shareholders on whose behalf they are taking decisions, which adds to the difficulties). Examples. Standard gambling comparisons---find your own examples. This method of estimating utilities suffers from the fact that it relies upon the decision-makers ability to answer hypothetical questions in the same way in which they would answer real ones. DIFFERING DEGREES OF RISK AVERSION Different people have different degrees of risk aversion. Graphically this is reflected in steeper or flatter Indifference curves in the risk return space. In figure below, we show a person with a relatively high degree of risk aversion contrasted with a person whose preferences indicate a relatively low degree of risk aversion. Points A and D are the same on both graphs. Project D is inferior to project A, since for the same expected return, E 0, it has the larger risk, R 1. In both cases the person would accept the risk level R 1 only if this is accompanied by an expected profit larger than that of project A. The more risk averse person in the left hand graph requires DB dollars in order to remain at the same level of utility and thus has a relatively high MRS of return for risk, measured by the ratio BD/AD. The less risk averse person on the right hand side requires only the considerably smaller amount of extra expected profit, DC dollars, for extra risk, R 1 R 2, and thus exhibits a relatively low MRS of return for risk measured by the ratio CD/AD.

8 RISK PREFERENCE AND RISK NEUTRALITY Risk preference and risk neutrality are not common among business decision makers. Consumers, on the other hand, may show risk preference or neutrality in such situations as gambling, sporting, and recreational activities. Risk preference means that risk is viewed as a utility producing good, and so the individual s indifference curves are negatively sloping as in the graph below. Such an individual is prepared to give up expected profits for a larger amount of risk. For example, a gambler might prefer a game in which the risk is greater and the expected value of gains is lower, over a safer bet on another game in which the expected value is somewhat higher.

9 RISK NEUTRALITY means that the individual is indifferent to risk, receiving neither utility nor disutility from risk regardless of the amount of risk involved. Such an individual s indifference curves would be horizontal. The arrow shows the direction of preference more expected profit is preferred to less, regardless of the risk. Consider an athlete who desperately wants to win the final game of the season. This individual will do whatever is necessary to help his team win or prevent the opposition from winning. Thus, hockey players block shots on goal with their faces and bodies, football players make suicidal plays that could easily result in broken bones, and racing drivers attempt that final pass on the last turn before the checkered flag.

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11 TECHNIQUES OF COPING WITH UNCERTAINTY The problems considered above all related to situations of risk, where the probabilities of different outcomes are known. If the probabilities are not known, the situation is one of uncertainty, rather than risk, and the techniques outlined above cannot be applied. Nevertheless, there are a number of different strategies that may be adopted in order to make decisions on rational criteria. THE MINIMAX CRITERION A firm making a decision may be characterized as choosing between alternative courses of action, whose outcomes depend upon which state of nature happens to be in force at the time of the action. In a situation of uncertainty the probabilities of the different states of nature are not known. In order to begin analyzing the problem a PAY-OFF MATRIX may be constructed as shown below. Each cell in the matrix shows the pay-off, which could be in terms of monetary values or utilities, for a given course of action, given the state of nature indicated. If the minimax criterion is adopted, the decision maker examines the worst pay-off for each action and then chooses the action for which the worst pay-off is highest. In the example given above the worst payoffs are as follows: In this example action 3 is selected, guaranteeing that the lowest payoff that will be received is 60. The minimax rule ensures that the worst possible outcomes are avoided and may be described as a pessimistic, conservative or high risk averse strategy. The obvious problem is

12 that it ignores the higher value pay-offs which may imply foregoing some possibly very large gains. The minimax regret criterion In the case of the minimax regret criterion the decision-maker considers the extent of the sacrifice made if a particular state of nature occurred but the best action for that state of nature was not chosen. In the example above, for state of nature A, action 1 involves a regret of 40, action 2 a regret of 100 and action 3 a regret of zero (0). Table below shows a complete regret matrix for all actions and states of nature.

13 Having set out the regret matrix, the action is chosen for which the largest regret is a minimum, leading to the choice of action 1. Such a strategy ensures that the maximum regret is not experienced, and is also a relatively pessimistic basis on which to make a decision. As in the case of the minimax criterion, the major criticism of this technique is that it only makes use of a very limited amount of information that is available, ignoring everything else. Actions that are rejected may have much smaller regrets than the one that is chosen, apart from their largest regrets. It is also possible that use of this approach could lead to inconsistent decisions in that if an action is chosen from a group of alternatives and then one of the rejected actions is deleted from the options, a different action may now be chosen, despite the fact that the original BEST option is still available. The maximax criterion This criterion is the opposite of the MINIMAX CRITERION in that the best outcomes of each action are identified, and then the action is selected for which the best outcome is largest. In the case set out in the first table this would lead to the choice of action 2. This is clearly an optimistic criterion to use in that it selects the action that provides a possibility of making the highest possible return. As in the case of the other criteria considered, its main failing is that it only takes a limited amount of the available information into account. The action that offers a prospect of the highest possible return may also be the one that offers the prospect of the highest possible loss (as in the example), but this is ignored. The Hurwicz alpha criterion The Hurwicz approach is an attempt to use more of the information available by constructing an index (the alpha index) for each action, which takes into account both the best and the worst outcomes and the extent to which the decision maker wishes to adopt a pessimistic or optimistic posture. The index is defined in the following way for each action:

14 I i = al i + (1-a) L i where: I i = the index for action I a= an optimism/pessimism index li = the lowest pay-off for action I L i = the highest pay-off for action I The action that has the largest alpha index is the one selected. The optimism/pessimism index may vary between 0 and 1 and may be estimated by increasing the value of x within the range for the situation shown in the table below until the decision maker is indifferent between the two actions. A very pessimistic decision-maker will be indifferent between the two actions at a very low value of x (they will be quite happy with a small certain gain compared with the prospect of either 0 or 1, because their pessimism leads them to suspect that the outcome of action 1 would be zero). On the other hand a very optimistic decision-maker who suspects that the outcome of action 1 is likely to be 1 will only be equally content with a certain amount that is almost as large as 1. Once the value of x has been estimated, through direct questioning of the decision maker, it is assumed that, as the decision-maker is indifferent between action 1 and action 2 they both have the same alpha index. From the formula given, action 1 has an index of (1-a) and action 2 has an index of x. It follows therefore that: (1 a) = x and the value of a can be calculated from the known value of x arrived at by experiment.

15 It should be noted that if x has a value of 1, so that a takes the value of zero, this indicates that the decision maker is very optimistic and the alpha criterion is exactly the same as the maximax criterion. Similarly, if the decision maker is highly pessimistic, so that a takes the value 1 the alpha criterion is equivalent to using the minimax approach. The Hurwicz technique therefore has the rather elegant property of encompassing minimax and maximin as special cases, each at a different end of the spectrum that may vary from highly optimistic to highly pessimistic. As the technique also makes use of more information than either maximax or minimax it may be said to be more superior to either of them in that respect. Nevertheless, like those other techniques it also wastes some of the available information concerning the possible outcomes of actions, using only the information for the best and the worst outcomes. Combinations of different strategies There is no reason to suppose, of course that firms do or must adopt any singe criterion in taking decisions under uncertainty. They may adopt different strategies on different occasions, or may consciously combine different strategies in order to spread the risk associated with any single approach.